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Q2.6 Simple integrals. Obtain estimates for these integrals, for by first finding
asymptotic expansions of the integrand for each relevant size of x, retaining
the first two terms in each case. (These integrals can be evaluated exactly,
so you may wish to check your results against the expansions of the exact values.)

(a) (b)

(c) (d)

Q2.7 More integrals. See Q2.6; repeat for these integrals (but here you are not
expected to have available the exact values).

(a) (b)

(c) for (d) for

(e) (f)
Q2.8 An integral from thin aerofoil theory. An integral of the type that can appear in the
study of thin aerofoil theory (for the velocity components in the flow field) is

obtain the first terms in the asymptotic expansions of the integrand (for
with x away from the end-points, for each of: (a) away from x
and away from (b) (c) Hence find an
estimate for Repeat the calculations with and then with
for away from the end-points, and then with
respectively. Again, find
estimates for and for show that your asymptotic
approximations for satisfy the matching principle.
Q2.9 Regular expansions for differential equations. Find the first two terms in the
asymptotic expansions of the solutions of these equations, satisfying the given
boundary conditions. In each case you should use the asymptotic sequence
and you should confirm that your 2-term expansions are uniformly valid.
(You may wish to examine the nature of the general term, and hence produce
an argument that shows the uniform validity of the expansion to all orders in
(a)
(b)
(c)
(d)
(e)